Average Error: 58.5 → 3.5
Time: 52.9s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{a} + \frac{1}{b}\]
\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}
↓
\frac{1}{a} + \frac{1}{b}
double f(double a, double b, double eps) {
        double r7081548 = eps;
        double r7081549 = a;
        double r7081550 = b;
        double r7081551 = r7081549 + r7081550;
        double r7081552 = r7081551 * r7081548;
        double r7081553 = exp(r7081552);
        double r7081554 = 1.0;
        double r7081555 = r7081553 - r7081554;
        double r7081556 = r7081548 * r7081555;
        double r7081557 = r7081549 * r7081548;
        double r7081558 = exp(r7081557);
        double r7081559 = r7081558 - r7081554;
        double r7081560 = r7081550 * r7081548;
        double r7081561 = exp(r7081560);
        double r7081562 = r7081561 - r7081554;
        double r7081563 = r7081559 * r7081562;
        double r7081564 = r7081556 / r7081563;
        return r7081564;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r7081565 = 1.0;
        double r7081566 = a;
        double r7081567 = r7081565 / r7081566;
        double r7081568 = b;
        double r7081569 = r7081565 / r7081568;
        double r7081570 = r7081567 + r7081569;
        return r7081570;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.5
Target14.8
Herbie3.5
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.5

    \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

  2. Taylor expanded around 0 56.8

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\varepsilon \cdot b + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right)\right)\right)}}\]

  3. Simplified55.8

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\varepsilon \cdot b + \left(\frac{1}{2} \cdot \left(\left(\varepsilon \cdot b\right) \cdot \left(\varepsilon \cdot b\right)\right) + \frac{1}{6} \cdot \left(\varepsilon \cdot \left(b \cdot \left(\left(\varepsilon \cdot b\right) \cdot \left(\varepsilon \cdot b\right)\right)\right)\right)\right)\right)}}\]

  4. Taylor expanded around 0 3.5

    \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]

  5. Final simplification3.5

    \[\leadsto \frac{1}{a} + \frac{1}{b}\]

Reproduce

herbie shell --seed 2019112 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :pre (and (< -1 eps) (< eps 1))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))))